I had erroneously believed that piano pitch was directly related to string length. What string length is necessary to create the lowest pitch on an 88-key piano (the low A)? With string lengths varying from piano to piano, i.e., from baby grand all the way to concert grand, how is it that pitches stay the same?
The frequency of a vibrating string is primarily based on three factors:
- The sounding length (longer is lower, shorter is higher)
- The tension on the string (more tension is higher, less is lower)
- The mass of the string, normally based on a uniform density per unit length (higher mass is lower, lower mass is higher)
To make a shorter string (such as in an upright piano) sound the same fundamental frequency as a longer string (such as in a 9' grand piano), either the thickness of the string must be increased (which increases the density and the mass) or the tension must be decreased, and usually it's a bit of both.
Thicker strings are often stiffer and that creates more inharmonic partials, and lower tension is associated with other problems, so the best way to make a string sound lower is the make it longer, but it is not practical to make a piano from strings that are all the same density and tension, because the lowest strings would have to be ridiculously long. Nine feet is already a great demand on space for a single musical instrument, and of course those pianos are extremely expensive and difficult to move.
To a degree you're correct. The shortest strings on any piano tend to be for the high notes, and the longest strings produce the lowest notes. But that's not enough. The tension on piano strings doesn't vary tremendously from the high to the low, so apart from length of string, the mass comes into play, so to speak.
The highest strings are just a single strand, although there are usually three of them, whereas the lowest strings are a single strand that has other wire wrapped round, creating a much thicker, heavier string. This gets round the problem, as well as a lot of pianos being overstrung, where the low sounding strings go diagonally across the soundboard, thus making them as long as possible, so they don't need to be quite so thick, or floppy, as they would on older, cheaper (studio) pianos, which had strings that only stretched from top to bottom.
If you want to be precise, pitch is inversely, but not solely related to length. (Inversely rather than directly, because as length increases, pitch decreases.)
As you say, this becomes obvious when you consider a full concert grand plays the same notes as a mini-piano (barring a few extra low ones). And when you consider that we tune a piano by adjusting tension rather than by chopping bits off the length of the strings.
Pitch is related to length, weight and tension. In practice, of course, we vary all three in the design, adjust just the tension thereafter.